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Theorem trljat1 29044
Description: The value of a translation of an atom  P not under the fiducial co-atom  W, joined with trace. Equation above Lemma C in [Crawley] p. 112. Todo: shorten with atmod3i1 28742? (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
trljat.l  |-  .<_  =  ( le `  K )
trljat.j  |-  .\/  =  ( join `  K )
trljat.a  |-  A  =  ( Atoms `  K )
trljat.h  |-  H  =  ( LHyp `  K
)
trljat.t  |-  T  =  ( ( LTrn `  K
) `  W )
trljat.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trljat1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )

Proof of Theorem trljat1
StepHypRef Expression
1 trljat.l . . . 4  |-  .<_  =  ( le `  K )
2 trljat.j . . . 4  |-  .\/  =  ( join `  K )
3 eqid 2253 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
4 trljat.a . . . 4  |-  A  =  ( Atoms `  K )
5 trljat.h . . . 4  |-  H  =  ( LHyp `  K
)
6 trljat.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
7 trljat.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
81, 2, 3, 4, 5, 6, 7trlval2 29041 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) W ) )
98oveq1d 5725 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( R `  F )  .\/  P )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
10 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
11 hllat 28242 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
13 simp3l 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
14 eqid 2253 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
1514, 4atbase 28168 . . . 4  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1613, 15syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
1714, 5, 6, 7trlcl 29042 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  e.  (
Base `  K )
)
18173adant3 980 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  e.  (
Base `  K )
)
1914, 2latjcom 14009 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( R `  F )  e.  ( Base `  K
) )  ->  ( P  .\/  ( R `  F ) )  =  ( ( R `  F )  .\/  P
) )
2012, 16, 18, 19syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( ( R `  F
)  .\/  P )
)
2114, 5, 6ltrncl 29003 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2216, 21syld3an3 1232 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  (
Base `  K )
)
2314, 2latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( F `  P ) )  e.  ( Base `  K
) )
2412, 16, 22, 23syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  e.  (
Base `  K )
)
25 simp1r 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
2614, 5lhpbase 28876 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2725, 26syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
2814, 1, 2latlej1 14010 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( F `  P )  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  ( F `  P )
) )
2912, 16, 22, 28syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  .<_  ( P  .\/  ( F `
 P ) ) )
3014, 1, 2, 3, 4atmod2i1 28739 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  ( F `  P )
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  ( F `  P )
) )  ->  (
( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
3110, 13, 24, 27, 29, 30syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( (
( P  .\/  ( F `  P )
) ( meet `  K
) W )  .\/  P )  =  ( ( P  .\/  ( F `
 P ) ) ( meet `  K
) ( W  .\/  P ) ) )
32 eqid 2253 . . . . . 6  |-  ( 1.
`  K )  =  ( 1. `  K
)
331, 2, 32, 4, 5lhpjat1 28898 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( W  .\/  P
)  =  ( 1.
`  K ) )
34333adant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  ( 1. `  K ) )
3534oveq2d 5726 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( W  .\/  P
) )  =  ( ( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) ) )
36 hlol 28240 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OL )
3710, 36syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OL )
3814, 3, 32olm11 28106 . . . 4  |-  ( ( K  e.  OL  /\  ( P  .\/  ( F `
 P ) )  e.  ( Base `  K
) )  ->  (
( P  .\/  ( F `  P )
) ( meet `  K
) ( 1. `  K ) )  =  ( P  .\/  ( F `  P )
) )
3937, 24, 38syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( P  .\/  ( F `  P ) ) (
meet `  K )
( 1. `  K
) )  =  ( P  .\/  ( F `
 P ) ) )
4031, 35, 393eqtrrd 2290 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( F `  P
) )  =  ( ( ( P  .\/  ( F `  P ) ) ( meet `  K
) W )  .\/  P ) )
419, 20, 403eqtr4d 2295 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  ( R `  F
) )  =  ( P  .\/  ( F `
 P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   1.cp1 13988   Latclat 13995   OLcol 28053   Atomscatm 28142   HLchlt 28229   LHypclh 28862   LTrncltrn 28979   trLctrl 29036
This theorem is referenced by:  trljat3  29046  trlval4  29066  trlval5  29067  cdlemc5  29073  cdlemk1  29709  cdlemk8  29716  cdlemki  29719  cdlemksv2  29725  cdlemk7  29726  cdlemk12  29728  cdlemk15  29733  cdlemk7u  29748  cdlemk12u  29750  cdlemk21N  29751  cdlemk20  29752  cdlemk22  29771  cdlemm10N  29997
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-padd 28674  df-lhyp 28866  df-laut 28867  df-ldil 28982  df-ltrn 28983  df-trl 29037
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